
For half-integer superspin, the relevant massless action is second-order and takes the form [1]}
\({\mathbb {S}}_{\rm {SO}}^{(2s+1)}[H,Y]&=& \bigg (- \frac{1}{2}\bigg )^s \int {\rm d}^{3|2} z \, E\,\bigg \lbrace -\frac{{\rm i}}{2} H^{\alpha (2s+1)} {\mathbb {Q}} H_{\alpha (2s+1)}-\frac{{\rm i}}{8} {\cal D}_\beta H^{\beta \alpha (2s)} {\cal D}^2 {\cal D}^\gamma H_{\gamma \alpha (2s)} \nonumber \\&&+ \frac{{\rm i}}{4} s {\cal D}_{\beta \gamma } H^{\beta \gamma \alpha (2s-1)}{\cal D}^{\rho \lambda } H_{\rho \lambda \alpha (2s-1)}-\frac{1}{2}(2s-1) Y^{\alpha (2s-2)} {\cal D}^{\beta \gamma } {\cal D}^\delta H_{\beta \gamma \delta \alpha (2s-2)} \nonumber \\&&+\frac{{\rm i}}{2} (2s-1)\Big [ Y^{\alpha (2s-2)} {\cal D}^2Y_{\alpha (2s-2)}- \frac{s-1}{s} {\cal D}_\beta Y^{\beta \alpha (2s-3)} {\cal D}^\gamma Y_{\gamma \alpha (2s-3)}\Big ]\nonumber \\&& +{\rm i}s {\cal S}H^{\beta \alpha (2s)} {\cal D}_\beta {}^\gamma H_{\gamma \alpha (2s)}+\frac{1}{2}(s+1) {{\cal S}} H^{\alpha (2s+1)} {\cal D}^2 H_{\alpha (2s+1)} \\&&+{\rm i}s (2s-3) {{\cal S}}^2 H^{\alpha (2s+1)} H_{\alpha (2s+1)}+ \frac{(2s-1)(s^2 -3s -2)}{s} {{\cal S}} Y^{\alpha (2s-2)} Y_{\alpha (2s-2)}\bigg \rbrace ~,~~~\nonumber \) 
